<p>In this paper, we always assume that <Emphasis Type="BoldItalic">q</Emphasis> is a power of an odd prime and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_811_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\( \mathbb {F}_q \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> is a finite field with <Emphasis Type="BoldItalic">q</Emphasis> elements. We determine all generator polynomials of cyclic expurgated or extended Goppa codes under some permutations induced by projective general linear matrices <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_811_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">\( \varvec{\widetilde{A}} \in \varvec{PGL_2(\mathbb {F}_q)} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mover accent="true"> <mi mathvariant="bold-italic">A</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> <mo>∈</mo> <mrow> <mi mathvariant="bold-italic">P</mi> <mi mathvariant="bold-italic">G</mi> <msub> <mi mathvariant="bold-italic">L</mi> <mn mathvariant="bold">2</mn> </msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi mathvariant="bold-italic">q</mi> </msub> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>